1. Field of the Invention
The invention refers to a digital simulation method of a non-linear interaction between an excitation source and a wave in a resonator and may be applied, in particular, to the digital synthesis, in real time, of an oscillating phenomenon such as the sound emitted by a musical instrument operating more particularly with sustained oscillations, such as wind or rubbed string instrument.
2. Description of the Related Art
The phenomena of wave propagation and formation of the emitted sounds, in particular, by a musical instrument have been studied scientifically for a very long time.
In particular, it is admitted, generally speaking, that a musical instrument includes, at least, one exciter, characterised by a non-linear characteristic, coupled possibly with certain linear elements (the reed, the lips, the bow, the hammer, etc. . . . ) and resonator elements, generally linear, where there is wave propagation as well as, generally, localised elements (for instance lateral bores or simple elements of the mass or spring type), generally linear as well.
Similarly, a digital instrument capable of synthesising the sounds emitted by a musical instrument, is composed generally of three main elements, respectively a first element, to sense the gests of a musician and to transform them into signals/control parameters, a second element computing the signal in real time, a third element converting this series of numbers calculated into a sound signal by means of digital/analogue converters, amplifier, loudspeakers.
The present invention concerns mainly the second real time calculation element of the signal.
It is known that the digital simulation of a sound or, more generally, of an oscillating phenomenon, may be conducted by discretisation in the time domain of equations forming the mathematic representation of the physical phenomenon to be simulated. Such a model is always expressed in the form of a system of equations with coupled partial derivations, linear or non-linear.
The simulation then consists, generally speaking, in computing as quickly as possible the solution of the acoustic/mechanical model describing the operation of the instrument or, at least, approximations preserving its most important characteristics.
Numerous methods exist to that effect and it is possible, in particular, to mention modal methods (which describe the resonator as a resonant filter comprised of a sum of elementary resonances), particular methods (which describe the medium wherein there is a wave propagation in the form of chains of the type mass-springs-dampers), or the digital methods for solving equations with partial derivations.
However, real time sound synthesis is difficult to realise and consequently, for some years, other methods have been developed, based on a “signal processing” formalism of the propagation in both directions of the resonator of the instrument. One may quote, for instance the methods called “digital wave guide” or “digital wave filter”.
Generally speaking, to represent the propagation of a wave, one may use, in the simplest formalism, the well known d'Alembert's equation, which applies to longitudinal waves (acoustic for instance) as well as transversal waves (vibration of a string for instance). In particular, in the case of the propagation of an acoustic wave, the acoustic pressure in all points of the resonator of a wind instrument may be split into a sum of two waves of acoustic pressure, one propagating from the player to the horn, and the other from the horn to the player, which are called away-wave and return-wave.
In practice, such propagation is expressed by a convolution equation (linear filtering), which yields the away wave (or return wave) at one point of the resonator at each time in relation to the away wave (or return wave) at another point at each instant. In the so-called Green formulation, which may be implemented in digital form, d'Alembert's equation specifies for instance that this linear filter, called Green core, is a pure delay, depending on the speed of propagation in the medium and on its length.
In a synthesis model, these waves, respectively away and return waves, are represented by two signals corresponding respectively to both propagative solutions of the differential equation.
Such a synthesis method is implemented, for instance in the document U.S. Pat. No. 5,332,862 which describes a synthesiser comprising generally speaking:                one non-linear part, simulating the exciter, to which two control parameters of the sound to be simulated are applied. These parameters are, in this case, the pressure of the player's breath and the pressure of his lips on the reed or the mouthpiece,        a linear part, simulating the resonator, which receives a signal noted q0, representative of the away wave, emitted by the non-linear part, and which emits a signal noted q1, representative of the return wave towards said part,        a means for creating the sound from the signals derived from the linear part and the non-linear part,        a digital/analogue converter generating the synthesised sound.        
Obviously, there are other types of synthesisers but, until now, all the methods involving a modelization the physical phenomena within the instrument were based on the decomposition of the vibration inside the resonator in terms of away wave and return wave variables.
Still, it has appeared that such methods exhibited several shortcomings.
First of all, when the acoustic resonator is formed, for instance, of several parts of cylindrical tubes of different diameters, the section change causes the generation of a transmitted wave and of a reflected wave at each interface. The phenomenon has been taken into account for a long time, for instance, within the framework of the modelization of the vocal conduit.
This type of modelization, which is identical, in its approach, to the conventional theory of the geometrical optics, is also employed, for instance, in seismic-reflection, in order to describe the propagation of elastic waves in a multilayer ground.
It is known, indeed that it is interesting, in all the cases when one or several waves propagate, to characterise an interface by a diffusion matrix, since it is thus possible to access the reflexions and transmissions of the different waves directly. However, the behaviour of this localised element often becomes difficult to grasp and to calculate, insofar as the interface and continuity equations are always written initially with physical quantities, for instance by expressing, at the interface, the continuity of the pressure or of the flow, of the strength or of the speed.
It is hence often more advantageous to use “impedance” or “admittance” matrices, which link the physical quantities directly, as described in an article of J. Kergomard, “Calculation of discontinuities in waveguides using mode-matching method: an alternative to the scattering matrix approach” J. d'acoustique 4 pp 111-138, (1991)”.
On the other hand, when there are localised elements other than interfaces in the instrument to be simulated, the “waveguide” method must be complemented by a “wave filter” type method describing such localised elements (such as mass, spring, dampers) for a correct connection between the various sub-systems.
Similarly, when the acoustic resonator is composed for instance of a conical pipe, the waves moving from the player to the horn (away waves) and from the horn to the player (return waves) are different, which requires, either different modelization, by means of two linear filters corresponding to the Green cores describing the propagation in each direction, or approximating the cone by a succession of short length cylinders of different diameters.
Besides, the sound produced by a musical instrument is not derived solely from the propagation of a wave through a resonator, regardless of the complexity of its geometry, but results from the non-linear coupling between said resonator and an excitation source. Such non-linear coupling is expressed physically between the physical quantities representing a cause (pressure in the acoustic case, strength in the mechanical case) and an effect (flow in the acoustic case, speed in the mechanical case), called Kirchhoff variables. In the acoustic case, this so-called Euler-Bernoulli physical law, a simplified version of the Navier-Stokes equations used in fluid mechanics, specifies that the acoustic pressure, at the reed or the lips of a wind instrument, is proportional, within one additive constant, to the square of the acoustic flow. The formulation of the waves in the resonator in the form of away waves and return waves requires therefore changing the variables enabling the non-linear coupling to be expressed, not in relation to the pressure-flow physical variables any longer, but in relation to these new away-wave and return-wave variables, as for instance, in the aforementioned document U.S. Pat. No. 5,332,862. Still, the variable change introduces an additional complexity in the synthesis method. It is thus that the use of iterative or tabulation methods has been recently suggested to calculate the solution of the non-linear system.
Consequently, the synthesis methods used until now do not enable to simply express the non-linear coupling which exists between the excitation source and the resonator of the instrument and limit the physical parameterisation of the synthesis algorithms.